Download Introduction to the Quantitative Reasoning Measure

GRADUATE RECORD EXAMINATIONS®
Introduction to the
Quantitative Reasoning Measure
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and other countries.
Introduction to the Quantitative
Reasoning Measure
The Quantitative Reasoning measure of the GRE revised General
Test assesses your:
• basic mathematical skills
• understanding of elementary mathematical concepts
• ability to reason quantitatively and to model and solve
problems with quantitative methods
Some of the questions in the measure are posed in real-life settings,
while others are posed in purely mathematical settings. The skills,
concepts, and abilities are tested in the four content areas below:
Arithmetic topics include properties and types of integers, such as
divisibility, factorization, prime numbers, remainders, and odd and
even integers; arithmetic operations, exponents, and radicals; and
concepts such as estimation, percent, ratio, rate, absolute value, the
number line, decimal representation, and sequences of numbers.
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Algebra topics include operations with exponents; factoring and
simplifying algebraic expressions; relations, functions, equations,
and inequalities; solving linear and quadratic equations and
inequalities; solving simultaneous equations and inequalities; setting
up equations to solve word problems; and coordinate geometry,
including graphs of functions, equations, and inequalities, intercepts,
and slopes of lines.
Geometry topics include parallel and perpendicular lines, circles,
triangles—including isosceles, equilateral, and 30∞ - 60∞ - 90∞
triangles—quadrilaterals, other polygons, congruent and similar
figures, three-dimensional figures, area, perimeter, volume, the
Pythagorean theorem, and angle measurement in degrees. The
ability to construct proofs is not tested.
Data analysis topics include basic descriptive statistics, such as
mean, median, mode, range, standard deviation, interquartile range,
quartiles, and percentiles; interpretation of data in tables and graphs,
such as line graphs, bar graphs, circle graphs, boxplots, scatterplots,
and frequency distributions; elementary probability, such as
probabilities of compound events and independent events; random
variables and probability distributions, including normal
distributions; and counting methods, such as combinations,
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permutations, and Venn diagrams. These topics are typically taught
in high school algebra courses or introductory statistics courses.
Inferential statistics is not tested.
The content in these areas includes high school mathematics and
statistics at a level that is generally no higher than a second course in
algebra; it does not include trigonometry, calculus, or other higherlevel mathematics. The publication Math Review for the GRE
Revised General Test provides detailed information about the
content of the Quantitative Reasoning measure.
The mathematical symbols, terminology, and conventions used in
the Quantitative Reasoning section are those that are standard at the
high school level. For example, the positive direction of a number
line is to the right, distances are nonnegative, and prime numbers are
greater than 1. Whenever nonstandard notation is used in a question,
it is explicitly introduced in the question.
In addition to conventions, there are some assumptions about
numbers and geometric figures that are used in the Quantitative
Reasoning measure. Two of these assumptions are (1) all numbers
used are real numbers and (2) geometric figures are not necessarily
drawn to scale. More about conventions and assumptions appears in
the publication Mathematical Conventions for the GRE Revised
General Test.
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Quantitative Reasoning Question Types
The Quantitative Reasoning section has four types of questions:
1.
2.
3.
4.
Quantitative Comparison
Multiple-choice—Select One
Multiple-choice—Select One or More
Numeric Entry
Each question appears either independently as a discrete question or
as part of a set of questions called a Data Interpretation set. All of
the questions in a Data Interpretation set are based on the same data
presented in tables, graphs, or other displays of data.
You are allowed to use a basic calculator on the Quantitative
Reasoning measure of the test. In the standard computer-based
version of the test, a basic calculator is provided on-screen.
In other editions of the test, a handheld basic calculator is
provided. No other calculator may be used except as an approved
accommodation. General information about using a calculator and
specific information on using the handheld basic calculator start on
page 68. Information about using the on-screen calculator in the
standard computer-based version of the test appears in Appendix B
of this document.
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Quantitative Comparison Questions
Description
Questions of this type ask you to compare two quantities—
Quantity A and Quantity B—and then determine which of
the following statements describes the comparison.
A.
B.
C.
D.
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
Tips for Answering
1. Become familiar with the answer choices.
Quantitative Comparison questions always have the same answer
choices, so get to know them, especially the last answer choice,
“The relationship cannot be determined from the information
given.” Never select this last choice if it is clear that the values
of the two quantities can be determined by computation. Also,
if you determine that one quantity is greater than the other, make
sure you carefully select the corresponding answer choice so as
not to reverse the first two answer choices.
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2. Avoid unnecessary computations.
Don’t waste time performing needless computations in order
to compare the two quantities. Simplify, transform, or estimate
one or both of the given quantities only as much as is necessary
to compare them.
3. Remember that geometric figures are not necessarily drawn
to scale.
If any aspect of a given geometric figure is not fully determined,
try to redraw the figure, keeping those aspects that are
completely determined by the given information fixed but
changing the aspects of the figure that are not determined.
Examine the results. What variations are possible in the relative
lengths of line segments or measures of angles?
4. Plug in numbers.
If one or both of the quantities are algebraic expressions, you
can substitute easy numbers for the variables and compare
the resulting quantities in your analysis. Consider all kinds of
appropriate numbers before you give an answer: for example,
zero, positive and negative numbers, small and large numbers,
fractions and decimals. If you see that Quantity A is greater
than Quantity B in one case and Quantity B is greater than
Quantity A in another case, choose “The relationship cannot
be determined from the information given.”
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5. Simplify the comparison.
If both quantities are algebraic or arithmetic expressions and
you cannot easily see a relationship between them, you can try
to simplify the comparison. Try a step-by-step simplification
that is similar to the steps involved when you solve the equation
5 = 4 x + 3 for x, or that is similar to the steps involved when
3y + 2
you determine that the inequality
< y is equivalent to
5
the simpler inequality 1 < y. Begin by setting up a comparison
involving the two quantities, as follows:
Quantity A
?
Quantity B
where ? is a “placeholder” that could represent the relationship
greater than (>), less than (<), or equal to (=) or could represent
the fact that the relationship cannot be determined from
the information given. Then try to simplify the comparison,
step-by-step, until you can determine a relationship between
simplified quantities. For example, you may conclude after
the last step that ? represents equal to (=). Based on this
conclusion, you may be able to compare Quantities A and B. To
understand this strategy more fully, see sample questions 6 to 9.
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Sample Questions
Directions: Compare Quantity A and Quantity B, using additional
information centered above the two quantities if such information
is given, and select one of the following four answer choices:
A.
B.
C.
D.
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
A symbol that appears more than once in a question has the same
meaning throughout the question.
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1.
A.
B.
C.
D.
Quantity A
Quantity B
(2)(6 )
2+6
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
Explanation
Since 12 is greater than 8, Quantity A is greater than
Quantity B. Thus, the correct answer is choice A,
Quantity A is greater.
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2.
Lionel is younger than Maria.
A.
B.
C.
D.
Quantity A
Quantity B
Twice Lionel’s age
Maria’s age
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
Explanation
If Lionel’s age is 6 years and Maria’s age is 10 years, then
Quantity A is greater, but if Lionel’s age is 4 years and
Maria’s age is 10 years, then Quantity B is greater. Thus,
the relationship cannot be determined. The correct answer
is choice D, the relationship cannot be determined from
the information given.
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3.
A.
B.
C.
D.
Quantity A
Quantity B
54% of 360
150
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
Explanation
In this question you are asked to compare 54% of 360 and 150.
Without doing the exact computation, you can see that 54
1
percent of 360 is greater than of 360, which is 180, and 180
2
is greater than Quantity B, 150. Thus, the correct answer is
choice A, Quantity A is greater.
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4.
Figure 1
A.
B.
C.
D.
Quantity A
Quantity B
The length of PS
The length of SR
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
Explanation
In this question you are asked to compare the length of PS
and the length of SR. From Figure 1, you know that PQR
is a triangle and that point S is between points P and R, so
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the length of PS is less than the length of PR and the length
of SR is less than the length of PR. You are also given that
the length of PQ is equal to the length of PR. However, this
information is not sufficient to compare the length of PS
and the length of SR. Furthermore, because the figure is not
necessarily drawn to scale, you cannot determine the relative
lengths of PS and SR from the figure, though the lengths may
appear to be equal. The position of S can vary along side PR
anywhere between P and R. Below are two possible variations
of Figure 1, each of which is drawn to be consistent with the
information that the length of PQ is equal to the length of PR.
Figure 2
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Figure 3
Note that Quantity A (the length of PS) is greater in Figure 2
and Quantity B (the length of SR) is greater in Figure 3.
Thus, the correct answer is choice D, the relationship
cannot be determined from the information given.
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5.
y = 2 x2 + 7x - 3
A.
B.
C.
D.
Quantity A
Quantity B
x
y
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
Explanation
In this question you are asked to compare x and y.
( )
If x = 0, then y = 2 02 + 7 (0 ) - 3 = -3, so in this case,
( )
x > y ; but if x = 1, then y = 2 12 + 7 (1) - 3 = 6, so in
that case, y > x. Thus, the correct answer is choice D, the
relationship cannot be determined from the information
given.
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Note that plugging numbers into expressions may not be conclusive.
It is conclusive, however, if you get different results after plugging
in different numbers: the conclusion is that the relationship cannot
be determined from the information given. It is also conclusive if
there are only a small number of possible numbers to plug in and
all of them yield the same result, say, that Quantity B is greater.
Now suppose that there are an infinite number of possible numbers
to plug in. If you plug many of them in and each time the result is,
for example, that Quantity A is greater, you still cannot conclude
that Quantity A is greater for every possible number that could be
plugged in. Further analysis would be necessary and should focus on
whether Quantity A is greater for all possible numbers or whether
there are numbers for which Quantity A is not greater.
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The following sample questions focus on simplifying
the comparison.
6.
y>4
A.
B.
C.
D.
Quantity A
Quantity B
3y + 2
5
y
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
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Explanation
Set up the initial comparison of Quantity A and Quantity B
using a placeholder symbol as follows:
3y + 2
5
?
y
Then simplify:
Step 1: Multiply both sides by 5 to get 3 y + 2
Step 2: Subtract 3y from both sides to get 2
Step 3: Divide both sides by 2 to get 1
?
?
?
5y
2y
y
The comparison is now simplified as much as possible. In order
to compare 1 and y, note that along with Quantities A and B
you were given the additional information y > 4. It follows
from y > 4 that y > 1, or 1 < y, so that in the comparison
1 ? y, the placeholder ? represents less than (<): 1 < y.
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However, the problem asks for a comparison between
Quantity A and Quantity B, not a comparison between 1
and y. To go from the comparison between 1 and y to
a comparison between Quantities A and B, start with the last
comparison, 1 < y, and carefully consider each simplification
step in reverse order to determine what each comparison
implies about the preceding comparison, all the way back to
the comparison between Quantities A and B if possible. Since
step 3 was “divide both sides by 2,” multiplying both sides of
the comparison 1 < y by 2 implies the preceding comparison
2 < 2 y, thus reversing step 3. Each simplification step can be
reversed as follows:
•
Step 3 was “Divide both sides by 2.” To reverse this step,
you need to multiply both sides by 2. The result of
reversing step 3 is 2 < 2 y .
•
Step 2 was “Subtract 3y from both sides.” To reverse
the step you need to add 3y to both sides. The result
of reversing step 2 is 3 y + 2 < 5 y .
•
Step 1 was “Multiply both sides by 5”. To reverse
this step divide both sides by 5. The result of reversing
3y + 2
step 1 is
< y.
5
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When each step is reversed, the relationship remains less than (<),
so Quantity A is less than Quantity B. Thus, the correct answer
is choice B, Quantity B is greater.
While some simplification steps like subtracting 3 from both sides
or dividing both sides by 10 are always reversible, it is important
to note that some steps, like squaring both sides, may not be
reversible.
Also, note that when you simplify an inequality, the steps of
multiplying or dividing both sides by a negative number change
the direction of the inequality; for example, if x < y, then - x > - y.
So the relationship in the final, simplified inequality may be the
opposite of the relationship between Quantities A and B. This
is another reason to consider the impact of each step carefully.
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7.
A.
B.
C.
D.
Quantity A
Quantity B
230 - 229
2
228
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
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Explanation
Set up the initial comparison of Quantity A and Quantity B:
230 - 229
2
?
228
Then simplify:
Step 1: Multiply both sides by 2 to get 230 - 229
Step 2: Add 229 to both sides to get 230
?
?
229
229 + 229
Step 3: Simplify the right-hand side using the fact that
(2 ) (229 ) = 230 to get 230
?
230
The resulting relationship is equal to (=). In reverse order, each
simplification step implies equal to in the preceding comparison.
So Quantities A and B are also equal. Thus, the correct answer
is choice C, the two quantities are equal.
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8.
A.
B.
C.
D.
Quantity A
Quantity B
x2 + 1
2x - 1
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
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Explanation
Set up the initial comparison of Quantity A and Quantity B:
x2 + 1
?
2x - 1
Then simplify by noting that the quadratic polynomial
x 2 - 2 x + 1 can be factored:
Step 1: Subtract 2x from both sides to get
x2 - 2 x + 1
?
-1
Step 2: Factor the left-hand side to get ( x - 1)
2
?
-1
The left-hand side of the comparison is the square of a number.
Since the square of a number is always greater than or equal
to 0, and 0 is greater than -1, the simplified comparison is the
inequality ( x - 1) > -1 and the resulting relationship is
2
greater than (>). In reverse order, each simplification step
implies the inequality greater than (>) in the preceding
comparison. Therefore, Quantity A is greater than Quantity B.
The correct answer is choice A, Quantity A is greater.
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9.
w >1
A.
B.
C.
D.
Quantity A
Quantity B
7w - 4
2w + 5
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the
information given.
Explanation
Set up the initial comparison of Quantity A and Quantity B:
7w - 4 ? 2 w + 5
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Then simplify:
Step 1: Subtract 2w from both sides and add 4 to both sides
to get 5w
?
9
Step 2: Divide both sides by 5 to get w
?
9
5
The comparison cannot be simplified any further. Although
you are given that w > 1, you still don’t know how w compares
9
to , or 1.8. For example, if w = 1.5, then w < 1.8, but if
5
w = 2, then w > 1.8. In other words, the relationship between
9
w and cannot be determined. Note that each of these
5
simplification steps is reversible, so in reverse order, each
simplification step implies that the relationship cannot be
determined in the preceding comparison. Thus, the relationship
between Quantities A and B cannot be determined. The correct
answer is choice D, the relationship cannot be determined
from the information given.
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The strategy of simplifying the comparison works most
efficiently when you note that a simplification step is reversible
while actually taking the step. Here are four common steps that
are always reversible:
1. Adding any number or expression to both sides
of a comparison
2. Subtracting any number or expression from both sides
3. Multiplying both sides by any nonzero number or
expression
4. Dividing both sides by any nonzero number or
expression
Remember that if the relationship is an inequality, multiplying
or dividing both sides by any negative number or expression
will yield the opposite inequality. Be aware that some common
operations like squaring both sides are generally not reversible
and may require further analysis using other information given
in the question in order to justify reversing such steps.
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Multiple-choice—Select One Questions
Description
These questions are multiple-choice questions that ask you to select
only one answer choice from a list of five choices.
Tips for Answering
1.
Use the fact that the answer is there.
If your answer is not one of the five answer choices given,
you should assume that your answer is incorrect and do
the following:
•
Reread the question carefully—you may have missed an
important detail or misinterpreted some information.
•
Check your computations—you may have made
a mistake, such as mis-keying a number on
the calculator.
•
Reevaluate your solution method—you may have
a flaw in your reasoning.
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2.
Examine the answer choices.
In some questions you are asked explicitly which of the choices
has a certain property. You may have to consider each choice
separately or you may be able to see a relationship between
the choices that will help you find the answer more quickly.
In other questions, it may be helpful to work backward from
the choices, say, by substituting the choices in an equation or
inequality to see which one works. However, be careful,
as that method may take more time than using reasoning.
3.
For questions that require approximations, scan the answer
choices to see how close an approximation is needed.
In other questions, too, it may be helpful to scan the choices
briefly before solving the problem to get a better sense of
what the question is asking. If computations are involved in
the solution, it may be necessary to carry out all computations
exactly and round only your final answer in order to get the
required degree of accuracy. In other questions, you may find
that estimation is sufficient and will help you avoid spending
time on long computations.
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Sample Questions
Directions: Select a single answer choice.
1. If 5 x + 32 = 4 - 2 x, what is the value of x ?
A. -4
B. -3
C.
4
D. 7
E. 12
Explanation
Solving the equation for x, you get 7 x = -28, and so x = -4.
The correct answer is choice A, - 4 .
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2. Which of the following numbers is farthest from the number 1
on the number line?
A. -10
B. -5
C.
0
D.
5
E. 10
Explanation
Circling each of the answer choices in a sketch of the number
line, as shown in Figure 4, shows that of the given numbers,
-10 is the greatest distance from 1.
Figure 4
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Another way to answer the question is to remember that the
distance between two numbers on the number line is equal
to the absolute value of the difference of the two numbers.
For example, the distance between -10 and 1 is -10 - 1 = 11,
and the distance between 10 and 1 is 10 - 1 = 9 = 9.
The correct answer is choice A, -10.
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3.
Figure 5
The figure above shows the graph of a function f, defined by
f ( x ) = 2 x + 4 for all numbers x. For which of the following
functions g defined for all numbers x does the graph of g
intersect the graph of f ?
A. g ( x ) = x - 2
B. g ( x ) = x + 3
C. g ( x ) = 2 x - 2
D. g ( x ) = 2 x + 3
E. g ( x ) = 3 x - 2
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Explanation
You can see that all five choices are linear functions whose
graphs are lines with various slopes and y-intercepts. The graph
of choice A is a line with slope 1 and y-intercept -2, shown in
Figure 6.
Figure 6
It is clear that this line will not intersect the graph of f to
the left of the y-axis. To the right of the y-axis, the graph
of f is a line with slope 2, which is greater than slope 1.
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Consequently, as the value of x increases, the value of y
increases faster for f than for g, and therefore the graphs do not
intersect to the right of the y-axis. Choice B is similarly ruled
out. Note that if the y-intercept of either of the lines in choices
A and B were greater than or equal to 4 instead of less than 4,
they would intersect the graph of f.
Choices C and D are lines with slope 2 and y-intercepts less
than 4. Hence, they are parallel to the graph of f (to the right
of the y-axis) and therefore will not intersect it. Any line with
a slope greater than 2 and a y-intercept less than 4, like the line
in choice E, will intersect the graph of f (to the right of the
y-axis). The correct answer is choice E, g ( x ) = 3 x - 2.
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4. A car got 33 miles per gallon using gasoline that cost $2.95
per gallon. What was the approximate cost, in dollars, of
the gasoline used in driving the car 350 miles?
A.
B.
C.
D.
E.
$10
$20
$30
$40
$50
Explanation
Scanning the answer choices indicates that you can do at least
some estimation and still answer confidently. The car used
350
350
gallons of gasoline, so the cost was
(2.95) dollars.
33
33
350
You can estimate the product
(2.95) by estimating 350
33
33
a little low, 10, and estimating 2.95 a little high, 3, to get
approximately (10 )(3) = 30 dollars. You can also use the
( )
( )
calculator to compute a more exact answer and then round
the answer to the nearest 10 dollars, as suggested by
the answer choices. The calculator yields the decimal
31.287. . . , which rounds to 30 dollars. Thus, the correct
answer is choice C, $30.
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5. A certain jar contains 60 jelly beans—22 white, 18 green, 11
yellow, 5 red, and 4 purple. If a jelly bean is to be chosen at
random, what is the probability that the jelly bean will be
neither red nor purple?
A.
B.
C.
D.
E.
0.09
0.15
0.54
0.85
0.91
Explanation
Since there are 5 red and 4 purple jelly beans in the jar, there
are 51 that are neither red nor purple and the probability of
51
selecting one of these is
. Since all of the answer choices
60
are decimals, you must convert the fraction to its decimal
equivalent, 0.85. Thus, the correct answer is choice D, 0.85.
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Multiple-choice—Select One or More Questions
Description
These questions are multiple-choice questions that ask you to select
one or more answer choices from a list of choices. A question may
or may not specify the number of choices to select.
Tips for Answering
1.
Note whether you are asked to indicate a specific number
of answer choices or all choices that apply.
In the latter case, be sure to consider all of the choices,
determine which ones are correct, and select all of those and
only those choices. Note that there may be only one correct
choice.
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2.
In some questions that involve inequalities that limit the
possible values of the answer choices, it may be efficient
to determine the least and/or the greatest possible value.
Knowing the least and/or greatest possible value may enable
you to quickly determine all of the choices that are correct.
3.
Avoid lengthy calculations by recognizing and continuing
numerical patterns.
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Sample Questions
Directions: Select one or more answer choices according to
the specific question directions.
If the question does not specify how many answer choices to select,
select all that apply.
•
The correct answer may be just one of the choices or may be
as many as all of the choices, depending on the question.
•
No credit is given unless you select all of the correct choices
and no others.
If the question specifies how many answer choices to select, select
exactly that number of choices.
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1. Which two of the following numbers have a product that is
greater than 60 ?
A. -9
B. -7
C. 6
D. 8
Explanation
For this type of question, it is often possible to exclude
some pairs of answer choices. In this question, the product
must be positive, so the only possible products are either
( -7)( -9) = 63 or (6 )(8) = 48. The correct answer
consists of choices A ( -9 ) and B ( -7 ).
-42-
2. Which of the following integers are multiples of both 2 and 3 ?
Indicate all such integers.
A.
B.
C.
D.
E.
F.
8
9
12
18
21
36
Explanation
You can first identify the multiples of 2, which are 8, 12, 18,
and 36, and then among the multiples of 2 identify the multiples
of 3, which are 12, 18, and 36. Alternatively, if you realize that
every number that is a multiple of 2 and 3 is also a multiple of
6, you can check which choices are multiples of 6. The correct
answer consists of choices C (12), D (18), and F (36).
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3. Each employee of a certain company is in either Department X
or Department Y, and there are more than twice as many
employees in Department X as in Department Y. The average
(arithmetic mean) salary is $25,000 for the employees
in Department X and is $35,000 for the employees in
Department Y. Which of the following amounts could be
the average salary for all of the employees in the company?
Indicate all such amounts.
A.
B.
C.
D.
E.
F.
G.
$26,000
$28,000
$29,000
$30,000
$31,000
$32,000
$34,000
Explanation
One strategy for answering this kind of question is to find
the least and/or greatest possible value. Clearly the average
salary is between $25,000 and $35,000, and all of the answer
choices are in this interval. Since you are told that there are
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more employees with the lower average salary, the average
salary of all employees must be less than the average of
$25,000 and $35,000, which is $30,000. If there were exactly
twice as many employees in Department X as in Department Y,
then the average salary for all employees would be, to the
nearest dollar, the following weighted mean,
(2 )(25,000 ) + (1)(35,000 )
2 +1
ª 28,333 dollars
where the weight for $25,000 is 2 and the weight for $35,000
is 1. Since there are more than twice as many employees in
Department X as in Department Y, the actual average salary
must be even closer to $25,000 because the weight for $25,000
is greater than 2. This means that $28,333 is the greatest
possible average. Among the choices given, the possible
values of the average are therefore $26,000 and $28,000.
Thus, the correct answer consists of choices A ($26,000)
and B ($28,000).
Intuitively, you might expect that any amount between $25,000
and $28,333 is a possible value of the average salary. To see
that $26,000 is possible, in the weighted mean above, use
the respective weights 9 and 1 instead of 2 and 1. To see that
$28,000 is possible, use the respective weights 7 and 3.
-45-
4. Which of the following could be the units digit of 57n,
where n is a positive integer?
Indicate all such digits.
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
0
1
2
3
4
5
6
7
8
9
-46-
Explanation
The units digit of 57n is the same as the units digit of 7n
for all positive integers n. To see why this is true for n = 2,
compute 572 by hand and observe how its units digit results
from the units digit of 72. Because this is true for every
positive integer n, you need to consider only powers of 7.
Beginning with n = 1 and proceeding consecutively, the units
digits of 7, 72 , 73 , 74 , and 75 are 7, 9, 3, 1, and 7,
respectively. In this sequence, the first digit, 7, appears again,
and the pattern of four digits, 7, 9, 3, 1, repeats without end.
Hence, these four digits are the only possible units digits of 7n
and therefore of 57n. The correct answer consists of choices
B (1), D (3), H (7), and J (9).
-47-
Numeric Entry Questions
Description
Questions of this type either ask you to enter your answer as an
integer or a decimal or ask you to enter your answer as a fraction.
In the standard computer-based version of the test, use the computer
mouse and keyboard to enter your answer. Detailed instructions for
entering answers in the standard computer-based version of the test
can be found in Appendix A of this document. Instructions for
entering answers in the large print, braille, and audio editions of
the test are included in corresponding Practice Test editions.
Tips for Answering
1.
Make sure you answer the question that is asked.
Since there are no answer choices to guide you, read the
question carefully and make sure you provide the type of
answer required. Sometimes there will be labels before
or after the answer space to indicate the appropriate type
of answer. Pay special attention to units such as feet or
miles, to orders of magnitude such as millions or billions,
and to percents as compared with decimals.
-48-
2.
If you are asked to round your answer, make sure you
round to the required degree of accuracy.
For example, if an answer of 46.7 is to be rounded to the nearest
integer, you need to enter the number 47. If your solution
strategy involves intermediate computations, you should carry
out all computations exactly and round only your final answer
in order to get the required degree of accuracy. If no rounding
instructions are given, enter the exact answer.
3.
Examine your answer to see if it is reasonable with respect
to the information given.
You may want to use estimation or another solution path
to double-check your answer.
-49-
Sample Questions
Directions: Enter your answer in the answer space below
the question.
Equivalent forms of the correct answer, such as 2.5 and 2.50, are all
correct. Fractions do not need to be reduced to lowest terms.
Enter the exact answer unless the question asks you to round your
answer.
1. One pen costs $0.25 and one marker costs $0.35. At those
prices, what is the total cost of 18 pens and 100 markers?
$
-50-
Explanation
Multiplying $0.25 by 18 yields $4.50, which is the cost of
the 18 pens; and multiplying $0.35 by 100 yields $35.00,
which is the cost of the 100 markers. The total cost is therefore
$4.50 + $35.00 = $39.50. Equivalent decimals, such as $39.5
or $39.500, are considered correct. Thus, the correct answer
is $39.50 (or equivalent).
Note that the dollar symbol is in front of the answer box, so the
symbol $ does not need to be entered in the box. In fact, only
numbers, a decimal point, and a negative sign can be entered
in the answer space.
-51-
2. Rectangle R has length 30 and width 10, and square S
has length 5. The perimeter of S is what fraction of
the perimeter of R ?
-52-
Explanation
The perimeter of R is 30 + 10 + 30 + 10 = 80, and the
perimeter of S is ( 4 )(5) = 20. Therefore, the perimeter
20
20
,
of the perimeter of R. To enter the answer
80
80
you should enter the numerator 20 in the top box and the
denominator 80 in the bottom box. Because the fraction does
not need to be reduced to lowest terms, any fraction that is
20
is also considered correct, as long as it fits
equivalent to
80
2
1
in the boxes. For example, both of the fractions and
8
4
20
are considered correct. Thus, the correct answer is
80
(or equivalent).
of S is
-53-
3.
Results of a Used-Car Auction
Small Cars
Large Cars
Number of cars offered
32
23
Number of cars sold
16
20
Projected sales total for cars
offered (in thousands)
$70
$150
Actual sales total (in thousands)
$41
$120
Figure 7
For the large cars sold at an auction, which is summarized in the
table in figure 7 above, what was the average sale price per car?
$
-54-
Explanation
From the table in Figure 7, you see that the number of large
cars sold was 20 and the sales total for large cars was $120,000
(not $120). Thus the average sale price per car was
$120,000
= $6,000. The correct answer is $6,000
20
(or equivalent).
(Note that the comma in 6,000 will appear automatically in the
answer box in the standard computer-based version of the test.)
-55-
4. A merchant made a profit of $5 on the sale of a sweater
that cost the merchant $15. What is the profit expressed
as a percent of the merchant’s cost?
Give your answer to the nearest whole percent.
%
Explanation
The percent profit is
( )
5
(100 ) = 33.333. . . = 33.3 percent,
15
which is 33%, to the nearest whole percent. Thus, the correct
answer is 33% (or equivalent).
-56-
If you are taking the standard computer-based version of
the test, and you use the calculator and the Transfer Display
button, the number that will be transferred to the answer space
is 33.333333, which is incorrect since it is not given to the
nearest whole percent. You will need to adjust the number
in the answer box by deleting all of the digits to the right of
the decimal point (using the Backspace key).
Also, since you are asked to give the answer as a percent, the
decimal equivalent of 33 percent, which is 0.33, is incorrect.
The percent symbol next to the answer space indicates that the
form of the answer must be a percent. Entering 0.33 in the box
would erroneously give the answer 0.33%.
-57-
5. Working alone at its constant rate, machine A produces k car
parts in 10 minutes. Working alone at its constant rate,
machine B produces k car parts in 15 minutes. How many
minutes does it take machines A and B, working simultaneously
at their respective constant rates, to produce k car parts?
minutes
Explanation
Machine A produces
k
parts per minute, and machine B
10
k
parts per minute. So when the machines work
15
simultaneously, the rate at which the parts are produced
is the sum of these two rates, which is
k
k
1
1
25
k
+
=k
+
=k
= parts per minute.
10 15
10 15
150
6
To compute the time required to produce k parts at this rate,
k
k
= 6. Therefore,
divide the amount k by the rate to get
k
6
6
the correct answer is 6 minutes (or equivalent).
produces
(
) ( )
-58-
One way to check that the answer of 6 minutes is reasonable
is to observe that if the slower rate of machine B were the same
as machine A’s faster rate of k parts in 10 minutes, then the
two machines, working simultaneously, would take half the
time, or 5 minutes, to produce the k parts. So the answer has
to be greater than 5 minutes. Similarly, if the faster rate of
machine A were the same as machine B’s slower rate of k parts
in 15 minutes, then the two machines, would take half the time,
or 7.5 minutes, to produce the k parts. So the answer has to be
less than 7.5 minutes. Thus, the answer of 6 minutes is
reasonable compared to the lower estimate of 5 minutes and
the upper estimate of 7.5 minutes.
-59-
Data Interpretation Sets
Description
Data Interpretation questions are grouped together and refer to the
same table, graph, or other data presentation. These questions ask
you to interpret or analyze the given data. The types of questions
may be Multiple-choice (both types) or Numeric Entry.
Tips for Answering
1.
Scan the data presentation briefly to see what it is about,
but do not spend time studying all of the information
in detail. Focus on those aspects of the data that are necessary
to answer the questions. Pay attention to the axes and scales
of graphs; to the units of measurement or orders of magnitude
(such as billions) that are given in the titles, labels, and legends;
and to any notes that clarify the data.
-60-
2.
Bar graphs and circle graphs, as well as other graphical
displays of data, are drawn to scale, so you can read or
estimate data from such graphs. For example, you can use
the relative sizes of bars or sectors to compare the quantities
that they represent, but be aware of broken scales and of bars
that do not start at 0.
3.
The questions are to be answered only on the basis of
the data presented, everyday facts (such as the number
of days in a year), and your knowledge of mathematics.
Do not make use of specialized information you may recall
from other sources about the particular context on which the
questions are based unless the information can be derived from
the data presented.
-61-
Sample Questions
Directions: Questions 1–3 are based on the following data.
Annual Percent Change in Dollar Amount of Sales
at Five Retail Stores from 2006 to 2008
Store
Percent Change
from
2006 to 2007
Percent Change
from
2007 to 2008
P
+10
–10
Q
–20
+9
R
+5
+12
S
–7
–15
T
+17
–8
Figure 8
-62-
1. If the dollar amount of sales at Store P was $800,000 for 2006,
what was the dollar amount of sales at that store for 2008 ?
A.
B.
C.
D.
E.
$727,200
$792,000
$800,000
$880,000
$968,000
Explanation
According to the table in Figure 8, if the dollar amount of sales
at Store P was $800,000 for 2006, then it was 10 percent greater
for 2007, which is 110 percent of that amount, or $880,000. For
2008 the amount was 90 percent of $880,000, which is
$792,000. The correct answer is choice B, $792,000.
Note that an increase of 10 percent for one year and a decrease
of 10 percent for the following year does not result in the same
dollar amount as the original dollar amount because the base
that is used in computing the percents is $800,000 for the first
change but $880,000 for the second change.
-63-
2. At Store T, the dollar amount of sales for 2007 was
what percent of the dollar amount of sales for 2008 ?
Give your answer to the nearest 0.1 percent.
%
-64-
Explanation
If A is the dollar amount of sales at Store T for 2007, then
8 percent of A, or 0.08 A, is the amount of decrease from 2007
to 2008. Thus A - 0.08 A = 0.92 A is the dollar amount for
2008. Therefore, the desired percent can be obtained by
dividing A by 0.92 A, which equals
A
1
=
= 1.0869565. . . . Expressed as a percent and
0.92 A 0.92
rounded to the nearest 0.1 percent, this number is 108.7%.
Thus, the correct answer is 108.7% (or equivalent).
-65-
3. Which of the following statements must be true?
Indicate all such statements.
A. For 2008 the dollar amount of sales at Store R was
greater than that at each of the other four stores.
B. The dollar amount of sales at Store S for 2008 was
22 percent less than that for 2006.
C. The dollar amount of sales at Store R for 2008 was
more than 17 percent greater than that for 2006.
Explanation
For choice A, since the only data given in Figure 8 are percent
changes from year to year, there is no way to compare the
actual dollar amount of sales at the stores for 2008 or for any
other year. Even though Store R had the greatest percent
increase from 2006 to 2008, its actual dollar amount of sales
for 2008 may have been much smaller than that for any of the
other four stores, and therefore choice A is not necessarily true.
-66-
For choice B, even though the sum of the two percent
decreases would suggest a 22 percent decrease, the bases
of the percents are different. If B is the dollar amount of
sales at Store S for 2006, then the dollar amount for 2007
is 93 percent of B, or 0.93 B, and the dollar amount for 2008
is given by (0.85)(0.93) B, which is 0.7905 B. Note that this
represents a percent decrease of 100 - 79.05 = 20.95 percent,
which is less than 22 percent, and so choice B is not true.
For choice C, if C is the dollar amount of sales at Store R for
2006, then the dollar amount for 2007 is given by 1.05C and
the dollar amount for 2008 is given by (1.12 )(1.05) C , which
is 1.176C. Note that this represents a 17.6 percent increase,
which is greater than 17 percent, so choice C must be true.
Therefore, the correct answer consists of only choice C
(The dollar amount of sales at Store R for 2008 was more
than 17 percent greater than that for 2006).
-67-
Using the Calculator
You are allowed to use a basic calculator on the Quantitative
Reasoning section of the test. In the standard computer-based
version of the test, a basic calculator is provided on-screen. In other
editions of the test, a handheld basic calculator is provided. No other
calculator may be used except as an approved accommodation. This
section provides general information about using a basic calculator,
as well as specific information about using the handheld basic
calculator provided. Information about using the on-screen
calculator provided in the standard computer-based version
of the test appears in Appendix B.
Sometimes the computations you need to do to answer a question
in the Quantitative Reasoning measure are somewhat tedious or
time-consuming, like long division or square roots. For such
computations, you can use the calculator provided to you at the
test site.
-68-
Although the calculator can shorten the time it takes to perform
computations, keep in mind that the calculator provides results that
supplement, but do not replace, your knowledge of mathematics.
You must use your mathematical knowledge to determine whether
the calculator’s results are reasonable and how the results can be
used to answer a question.
Here are five general guidelines for calculator use in the
Quantitative Reasoning measure:
1. Most of the questions don’t require difficult computations,
so don’t use the calculator just because it’s available.
2. Use it for calculations that you know are tedious, such as
long division, square roots, and addition, subtraction, or
multiplication of numbers that have several digits.
-69-
3. Avoid using it for simple computations that are quicker
4,300
, 25,
to do mentally, such as 10 - 490, ( 4 )(70 ) ,
10
and 302.
4. Some questions can be answered more quickly by reasoning
and estimating than by using the calculator.
5. If you use the calculator, estimate the answer beforehand
so you can determine whether the calculator’s answer is
“in the ballpark.” This may help you avoid key-entry errors.
The following three guidelines are specific to the handheld
calculator provided with editions of the test other than the standard
computer-based version:
1. Some computations are not defined for real numbers;
for example, division by zero or taking the square root
of a negative number. The calculator will indicate that
these are errors.
2. The calculator displays up to eight digits. If a computation
results in a number larger than 99,999,999, then the
calculator will indicate that this is an error. For example, the
calculation 10,000,000 ¥ 10 = results in an error.
-70-
3. When a computation involves more than one operation,
the calculator performs the operations one by one in
the order in which they are entered. For example, when
the computation 1 + 2 ¥ 4 is entered into the calculator,
the result is 12. To get this result, the calculator adds
1 and 2, displays a result of 3, and then multiplies 3
and 4 and displays a result of 12. The calculator does not
perform operations with respect to the mathematical
convention called order of operations, described below.
The order of operations convention, which is purely
mathematical and predates calculators, establishes which
operations are performed before others in a mathematical
expression that has more than one operation. The order is
as follows: parentheses, exponentiation (including square
roots), multiplications and divisions (from left to right),
additions and subtractions (from left to right). For example,
the value of the expression 1 + 2 ¥ 4 calculated with respect
to order of operations is 9, because the expression is evaluated
by first multiplying 2 and 4 and then by adding 1 to the result.
-71-
Some calculators perform multiple operations using the order
of operations convention, but the handheld calculator on the
Quantitative Reasoning measure does not; again, it performs
multiple operations one by one in the order that they are entered
into the calculator.
Below is an example of a computation using the handheld
calculator.
Example: Compute 4 +
6.73
.
2
-72-
Explanation
Perform the division first; that is, enter 6.73 ∏ 2 =
3.365, and then enter
+ 4 =
to get
to get 7.365. Note that if you
enter 4 + 6.73 ∏ 2 = , the answer will be incorrect, because
the calculator would perform the addition before the division,
4 + 6.73
6.73
resulting in
.
rather than 4 +
2
2
-73-
Appendix A
Numeric Entry Directions for the Standard
Computer-Based Version of the Test
Directions: Enter your answer as an integer or a decimal if there
is a single answer box OR as a fraction if there are two separate
boxes—one for the numerator and one for the denominator.
To enter an integer or a decimal, either type the number in the
answer box using the keyboard or use the Transfer Display button
on the calculator.
•
First, click on the answer box—a cursor will appear
in the box —and then type the number.
•
To erase a number, use the Backspace key.
•
For a negative sign, type a hyphen. For a decimal point,
type a period.
•
To remove a negative sign, type the hyphen again and
it will disappear; the number will remain.
-74-
•
The Transfer Display button on the calculator will transfer
the calculator display to the answer box.
•
Equivalent forms of the correct answer, such as 2.5
and 2.50, are all correct.
•
Enter the exact answer unless the question asks you
to round your answer.
To enter a fraction, type the numerator and the denominator
in the respective boxes using the keyboard.
•
For a negative sign, type a hyphen. A decimal point cannot
be used in the fraction.
•
The Transfer Display button on the calculator cannot
be used for a fraction.
•
Fractions do not need to be reduced to lowest terms, though
you may need to reduce your fraction to fit in the boxes.
-75-
Appendix B
Using the on-Screen Calculator Provided
with the Standard Computer-Based Version
of the Test
The on-screen calculator provided with the standard computer-based
version of the test is shown in Figure 9.
Figure 9
-76-
The following seven guidelines are specific to the on-screen
calculator:
1. When you use the computer mouse or the keyboard
to operate the calculator, take care not to miskey
a number or operation.
2. Note all of the calculator’s buttons, including Transfer
Display.
3. The Transfer Display button can be used on Numeric Entry
questions with a single answer box. This button will transfer
the calculator display to the answer box. You should check
that the transferred number has the correct form to answer
the question. For example, if a question requires you to
round your answer or convert your answer to a percent, make
sure that you adjust the transferred number accordingly.
4. Take note that the calculator respects order of operations,
as explained below.
The order of operations convention, which is purely mathematical
and predates calculators, establishes which operations are performed
-77-
before others in a mathematical expression that has more than one
operation. The order is as follows: parentheses, exponentiation
(including square roots), multiplications and divisions (from left
to right), additions and subtractions (from left to right). With respect
to order of operations, the value of the expression 1 + 2 ¥ 4 is 9
because the expression is evaluated by first multiplying 2 and 4
and then by adding 1 to the result. This is how the calculator in
the Quantitative Reasoning measure performs the operations.
5. In addition to parentheses, the on-screen calculator has
one memory location and three memory buttons that
govern it: memory recall MR , memory clear MC ,
and memory sum M+ . These buttons function as they
normally do on most basic calculators.
6. Some computations are not defined for real numbers;
for example, division by zero or taking the square root
of a negative number. If you enter 6 ∏ 0 = , the word
Error will be displayed. Similarly, if you enter 1 ±
,
then Error will be displayed. To clear the display, you must
press the clear button C .
-78-
7. The calculator displays up to eight digits. If a computation
results in a number larger than 99,999,999, then Error
will be displayed. For example, the calculation
10,000,000 ¥ 10 = results in Error. The clear button
C must be used to clear the display.
Below are six examples of computations using the calculator.
Example 1. Compute 4 +
6.73
.
2
Explanation
Enter 4 + 6.73 ∏ 2 =
enter 6.73 ∏ 2 =
to get 7.365. Alternatively,
to get 3.365, and then enter
to get 7.365.
-79-
+ 4 =
Example 2. Compute -
8.4 + 9.3
.
70
Explanation
Since division takes precedence over addition in the order
of operations, you need to override that precedence in order
to compute this fraction. Here are two ways to do that.
You can use the parentheses for the addition in the numerator,
entering
( 8.4 + 9.3 )
∏ 70 = ± to get -0.2528571.
Or you can use the equals sign after 9.3, entering
8.4 + 9.3 = ∏ 70 = ± to get the same result. In the
second way, note that pressing the first = is essential, because
without it, 8.4 + 9.3 ∏ 70 = ± would erroneously
(
compute - 8.4 +
)
9.3
instead. Incidentally, the exact result of
70
the computation is the repeating decimal -0.25285714, where
the digits 285714 repeat without ending, but the calculator
rounds the decimal to -0.2528571.
-80-
Example 3. Find the length, to the nearest 0.01, of the
hypotenuse of a right triangle with legs of length 21 and 54;
that is, use the Pythagorean Theorem to calculate 212 + 542 .
Explanation
Enter 21 ¥ 21 + 54 ¥ 54 =
to get 57.939624.
Again, pressing the = before the
is essential because
21 ¥ 21 + 54 ¥ 54
=
would erroneously compute
212 + 54 54. This is because the square root would take
precedence over the division in the order of operations.
Note that parentheses could be used, as in
( 21 ¥ 21 )
+
( 54 ¥ 54 )
=
but they
are not necessary because the multiplications already take
precedence over the addition. Incidentally, the exact answer is
a nonterminating, nonrepeating decimal, or an irrational number,
but the calculator rounds the decimal to 57.939624. Finally,
note that the problem asks for the answer to the nearest 0.01,
so the correct answer is 57.94.
-81-
Example 4. Compute ( -15) .
3
Explanation
Enter 15 ± ¥ 15 ± ¥ 15 ± = to get -3, 375.
-82-
Example 5. Convert 6 miles per hour to feet per second.
Explanation
The solution to this problem uses the conversion factors
1 mile = 5,280 feet and 1 hour = 3,600 seconds as follows:
(
)(
)(
)
6 miles 5,280 feet
1 hour
feet
= ?
1 hour
1 mile
3,600 seconds
second
Enter 6 ¥ 5280 ∏ 3600 =
enter 6 ¥ 5280 =
enter
to get 8.8. Alternatively,
to get the result 31,680, and then
∏ 3600 = to get 8.8 feet per second.
-83-
Example 6. At a fund-raising event, 43 participants donated $60
each, 21 participants donated $80 each, and 16 participants
donated $100 each. What was the average (arithmetic mean)
donation per participant, in dollars?
Explanation
The solution to this problem is to compute the weighted mean
(43)(60 ) + (21)(80) + (16 )(100 )
. In using the calculator to
43 + 21 + 16
compute the weighted mean, it is useful to take advantage of
the calculator’s memory buttons and parenthesis.
To do this calculation using the calculator’s memory buttons
note the following:
1. To enter a number into memory, make sure the memory
is clear, then enter the number followed by the M+ button.
(Note that if there is a number in memory M will appear
to the left of the display. To clear the memory use
the MC button. When the memory is clear the M next to
the display will disappear.)
-84-
2. To add a number, n, to a number stored in memory, t,
enter the number n followed by the M+ button. At the end
of this computation the number n will appear in the display,
and the number n + t will be stored in memory. To retrieve
the number n + t from memory, use the MR button.
To compute the weighted mean first calculate the value of
the numerator and store it in memory as follows:
43 ¥ 60 = M+ 21 ¥ 80 = M+ 16 ¥ 100 = M+
Then retrieve the numerator from memory and divide it by the
denominator as follows: (Note that you need to put parenthesis
around the sum in the denominator.)
MR ∏
( 43 + 21 + 16 )
=
The result is 73.25, or $73.25 per participant.
When the MR button is pressed in the computation above,
the current value in memory, 5,860, is displayed.
-85-